Finite Group Databasebeta

← #18006092

Subgroups — #18006092

Order 4736.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924561C211yesmaximal, characteristic, complemented
2C22×#10596941C211yesmaximal, characteristic, complemented
3D4×D148C211yesmaximal, characteristic, complemented
4#14557923C211yesmaximal, characteristic, complemented
5#14557781C211yesmaximal, characteristic, complemented
6#14557767C211yesmaximal, characteristic, complemented
7#14557745C211yesmaximal, characteristic, complemented
8#14557733C211yesmaximal, characteristic, complemented
9#14557683C211yesmaximal, characteristic, complemented
10#14557678C211yesmaximal, characteristic, complemented
11#14557672C211yesmaximal, characteristic, complemented
12#14557666C211yesmaximal, characteristic, complemented
13#14557631C211yesmaximal, characteristic, complemented
14#14557605C211yesmaximal, characteristic, complemented
15C37×#548C211yesmaximal, characteristic, complemented
16C22×D148C2211yescharacteristic
17C22×D148C2211yescharacteristic
18C2×D4×D37—21no—
19#11924561—24no—
20C2×#10596941—24no—
21C2×#10596941C2211yescharacteristic
22C2×#10596941C2211yescharacteristic
23C2×#10596941C2211yescharacteristic, complemented
24C2×#10596941C2211yescharacteristic, complemented
25C2×#10596941C2211yescharacteristic, complemented
26C2×#10596941C2211yescharacteristic, complemented
27C2×#10596939C2211yescharacteristic
28C2×#10596939C2211yescharacteristic, complemented
29C2×#10596939C2211yescharacteristic, complemented
30C2×#10596934C2211yescharacteristic, complemented
31C22×#64436C2211yescharacteristic, complemented
32#11924533C2211yescharacteristic
33#11924511—21no—
34#11924510C2211yescharacteristic
35#11924510C2211yescharacteristic
36#11924506—21no—
37#11924506—21no—
38C4×D148C2211yescharacteristic
39#11924469C2211yescharacteristic, complemented
40C4⋊#64436C2211yescharacteristic, complemented
41#11924460C2211yescharacteristic, complemented
42#11924460C2211yescharacteristic, complemented
43#11924457C2211yescharacteristic, complemented
44#11924456C2211yescharacteristic
45#11924456C2211yescharacteristic, complemented
46#11924456C2211yescharacteristic, complemented
47C22⋊#64436C2211yescharacteristic, complemented
48C22⋊#64436C2211yescharacteristic, complemented
49#11924447C2211yescharacteristic, complemented
50C2×#41×C37C2211yescharacteristic, complemented
51C22×Q8×C37C2211yescharacteristic, complemented
52#124×C37—21no—
53#124×C37C2211yescharacteristic, complemented
54C22⋊Q8×C37—21no—
55C22⋊Q8×C37C2211yescharacteristic, complemented
56#121×C37—21no—
57#121×C37C2211yescharacteristic, complemented
58C4×Q8×C37C2211yescharacteristic, complemented
59C4×D4×C37—21no—
60C4×D4×C37C2211yescharacteristic
61C23×D37—21no—
62C23×D37—21no—
63#41×C37—22no—
64#41×C37—41no—
65C2×Q8×C37—21no—
66C2×Q8×C37—22no—
67C2×Q8×C37—41no—
68C2×Q8×C37C2311yescharacteristic
69C2×Q8×C37C2311yescharacteristic
70C2×D4×C37—21no—
71C2×D4×C37—21no—
72C2×D4×C37D411yescharacteristic
73C2×D4×C37C2311yescharacteristic
74C22×C4×C37—21no—
75C22×C4×C37—21no—
76C22×C4×C37C2311yescharacteristic
77C22×C4×C37C2311yescharacteristic
78C2×C22⋊D37—21no—
79C2×C22⋊D37—21no—
80D4×D37—44no—
81C2×D148—22no—
82C2×D148—22no—
83C2×D148—21no—
84C2×D148—21no—
85C2×D148—21no—
86C2×D148—21no—
87C2×D148—41no—
88C2×D148—41no—
89C2×D148C2311yescharacteristic
90C2×D148C2311yescharacteristic
91C2×D148C2311yescharacteristic
92C2×C4×D37—21no—
93C4⋊C4×C37—21no—
94C4⋊C4×C37—21no—
95C4⋊C4×C37D411yescharacteristic
96C4⋊C4×C37C2311yescharacteristic
97C22⋊C4×C37—21no—
98C22⋊C4×C37—21no—
99C22⋊C4×C37—21no—
100C22⋊C4×C37—21no—
101C22⋊C4×C37D411yescharacteristic
102C22⋊C4×C37D411yescharacteristic
103C42×C37—21no—
104C42×C37C2311yescharacteristic
105#10596941—22no—
106#10596941—22no—
107#10596941—22no—
108#10596941—22no—
109#10596941—41no—
110#10596941D414yescomplemented
111#10596939—22no—
112#10596939—22no—
113#10596939—41no—
114#10596938—21no—
115#10596938—21no—
116#10596938—21no—
117C4⋊Dic37C2311yescharacteristic
118#10596934—22no—
119C2×#64436—22no—
120C2×#64436C2311yescharacteristic
121C2×#64436C2311yescharacteristic
122C2×#64436C2311yescharacteristic
123C2×#64436C2311yescharacteristic
124C23×C37—21no—
125C23×C37C2×D411yescharacteristic
126C23×C37C2×D411yescharacteristic
127C22×D37—21no—
128C22×D37—21no—
129C22×D37—21no—
130C22×D37—42no—
131C22×D37—42no—
132C22×D37—42no—
133C22×D37—42no—
134C22×D37—81no—
135C22×D37—81no—
136Q8×C37—21no—
137Q8×C37—21no—
138Q8×C37—21no—
139Q8×C37—42no—
140Q8×C37C2×D412yescomplemented
141D4×C37—22no—
142D4×C37—21no—
143D4×C37—21no—
144D4×C37—42no—
145D4×C37—42no—
146C2×C4×C37—21no—
147C2×C4×C37—21no—
148C2×C4×C37—21no—
149C2×C4×C37—21no—
150C2×C4×C37—21no—
151C2×C4×C37—21no—
152C2×C4×C37—21no—
153C2×C4×C37—42no—
154C2×C4×C37—42no—
155C2×C4×C37C2×D411yescharacteristic
156C2×C4×C37C2×D411yescharacteristic
157C2×C4×C37C2×D411yescharacteristic
158C2×C4×C37C2×D411yescharacteristic
159C2×C4×C37C2411yescharacteristic
160C22⋊D37—42no—
161C22⋊D37—42no—
162C2×Dic37—21no—
163D148—21no—
164D148—21no—
165D148—21no—
166D148—21no—
167D148—21no—
168D148—42no—
169D148—42no—
170D148—42no—
171D148—81no—
172D148C2×D412yes—
173C4×D37—42no—
174#64436—21no—
175#64436—21no—
176#64436—21no—
177#64436C2×D412yescomplemented
178C22×C37—21no—
179C22×C37—21no—
180C22×C37—21no—
181C22×C37—21no—
182C22×C37—21no—
183C22×C37—41no—
184C22×C37—41no—
185C22×C37—41no—
186C22×C37AGL(1,ℤ/8)12yescomplemented
187C22×C37C22×D411yescharacteristic
188C2×D37—22no—
189C2×D37—41no—
190C2×D37—41no—
191C2×D37—41no—
192C2×D37—41no—
193C2×D37—41no—
194C2×D37—41no—
195C2×D37—41no—
196C2×D37—41no—
197C2×D37—84no—
198C2×D37—84no—
199C4×C37—21no—
200C4×C37—21no—
201C4×C37—22no—
202C4×C37—21no—
203C4×C37—21no—
204C4×C37—41no—
205C4×C37—41no—
206C4×C37C22×D411yescharacteristic
207C4×C37H(5,𝔽2)11yescharacteristic
208Dic37—41no—
209#3265—371nomaximal, Sylow
210C2×C37—22no—
211C2×C37—41no—
212C2×C37—81no—
213C2×C37D4211yescharacteristic
214C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C37#57711yescharacteristic
216D37—42no—
217D37—81no—
218D37—81no—
219C2×AGL(1,ℤ/8)—371no—
220C22×SD16—371no—
221#548D3711yescharacteristic, complemented
222D42—371no—
223#496—371no—
224#468—371no—
225#466—371no—
226#463—371no—
227#460—371no—
228#450—371no—
229#449—371no—
230#448—371no—
231#447—371no—
232#440—371no—
233#435—371no—
234C37#326511yescharacteristic, complemented, Sylow
235C2×#41C2×D3711yescharacteristic, complemented
236C22×Q8C2×D3711yescharacteristic, complemented
237C22×D4—371no—
238C22×D4—371no—
239C22×D4—741no—
240AGL(1,ℤ/8)—744no—
241C2×SD16—371no—
242C2×SD16—371no—
243C2×SD16—371no—
244C2×SD16—371no—
245C2×SD16—371no—
246C2×SD16—371no—
247C2×SD16—744no—
248C2×D8—371no—
249C2×D8—371no—
250C2×D8—371no—
251C2×M16—371no—
252C22×C8—371no—
253#127—371no—
254#124—21no—
255#124C2×D3711yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D3711yescharacteristic, complemented
258#121—21no—
259#121—371no—
260#121—371no—
261#121—741no—
262#121C2×D3711yescharacteristic, complemented
263#120—741no—
264#120—741no—
265C4×Q8C2×D3711yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—371no—
268C4×D4C2×D3711yescharacteristic
269#107—371no—
270C4⋊C8—371no—
271#103—371no—
272#103—371no—
273#102—371no—
274#102—371no—
275#102—371no—
276#102—371no—
277C22⋊C8—371no—
278C22⋊C8—371no—
279#97—371no—
280C24—741no—
281C24—741no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D3711yescharacteristic
288C2×Q8C22×D3711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—371no—
292C2×D4—371no—
293C2×D4—371no—
294C2×D4—742no—
295C2×D4—742no—
296C2×D4—741no—
297C2×D4—741no—
298C2×D4—741no—
299C2×D4—741no—
300C2×D4—741no—
301C2×D4—741no—
302C2×D4—1481no—
303C2×D4—1481no—
304C2×D4—1484no—
305C2×D4C22⋊D3711yescharacteristic
306C2×D4C22×D3711yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—741no—
310C22×C4C22×D3711yescharacteristic
311C22×C4C22×D3711yescharacteristic
312SD16—374no—
313SD16—742no—
314SD16—742no—
315SD16—742no—
316SD16—742no—
317SD16—1481no—
318D8—742no—
319D8—742no—
320D8—1481no—
321M16—742no—
322C2×C8—371no—
323C2×C8—371no—
324C2×C8—371no—
325C2×C8—371no—
326C2×C8—742no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—371no—
330C4⋊C4C22⋊D3711yescharacteristic
331C4⋊C4C22×D3711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—741no—
337C22⋊C4—741no—
338C22⋊C4—741no—
339C22⋊C4C22⋊D3711yescharacteristic
340C22⋊C4C22⋊D3711yescharacteristic
341C42—21no—
342C42C22×D3711yescharacteristic
343C23—21no—
344C23—2961no—
345C23—2961no—
346C23—741no—
347C23—741no—
348C23—741no—
349C23—1482no—
350C23—1482no—
351C23—1482no—
352C23—1482no—
353C23C2×C22⋊D3711yescharacteristic
354C23C2×C22⋊D3711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D3712yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—372no—
366D4—2961no—
367D4—741no—
368D4—741no—
369D4—741no—
370D4—741no—
371D4—741no—
372D4—1482no—
373D4—1482no—
374D4—1482no—
375D4—1482no—
376D4—1482no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—741no—
387C2×C4—1482no—
388C2×C4C2×C22⋊D3711yescharacteristic
389C2×C4C2×C22⋊D3711yescharacteristic
390C2×C4C2×C22⋊D3711yescharacteristic
391C2×C4C2×C22⋊D3711yescharacteristic
392C2×C4C23×D3711yescharacteristic
393C8—372no—
394C8—741no—
395C8—741no—
396C8—741no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—2964no—
406C22—2964no—
407C22—742no—
408C22—1481no—
409C22—1481no—
410C22—1481no—
411C22—1481no—
412C22—1481no—
413C22—1481no—
414C22—1481no—
415C22—1481no—
416C22#1192456112yescomplemented
417C22C22×C22⋊D3711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—1481no—
426C4C2×D4×D3711yescharacteristic
427C4#1192458111yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—2961no—
432C2—2961no—
433C2—1482no—
434C2D4×C22⋊D3711yescharacteristic
435C2C2×#1192456111yescharacteristic
436C2#1455834111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#548C2
socleC22×C37C22×D4
hypercenter#548D37
hypocenterC37#3265
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#11924561C2semidirect
2. C22×#10596941C2semidirect
3. D4×D148C2semidirect
4. #14557923C2semidirect
5. #14557781C2semidirect
6. #14557767C2semidirect
7. #14557745C2semidirect
8. #14557733C2semidirect
9. #14557683C2semidirect
10. #14557678C2semidirect
11. #14557672C2semidirect
12. #14557666C2semidirect
13. #14557631C2semidirect
14. #14557605C2semidirect
15. C37×#548C2semidirect
23. C2×#10596941C22semidirect
24. C2×#10596941C22semidirect
25. C2×#10596941C22semidirect
26. C2×#10596941C22semidirect
28. C2×#10596939C22semidirect
29. C2×#10596939C22semidirect
30. C2×#10596934C22semidirect
31. C22×#64436C22semidirect
39. #11924469C22semidirect
40. C4⋊#64436C22semidirect
41. #11924460C22semidirect
42. #11924460C22semidirect
43. #11924457C22semidirect
45. #11924456C22semidirect
46. #11924456C22semidirect
47. C22⋊#64436C22semidirect
48. C22⋊#64436C22semidirect
49. #11924447C22semidirect
50. C2×#41×C37C22semidirect
51. C22×Q8×C37C22semidirect
53. #124×C37C22semidirect
55. C22⋊Q8×C37C22semidirect
57. #121×C37C22semidirect
58. C4×Q8×C37C22semidirect
110. #10596941D4semidirect
140. Q8×C37C2×D4semidirect
177. #64436C2×D4semidirect
186. C22×C37AGL(1,ℤ/8)semidirect
221. #548D37semidirect
234. C37#3265semidirect
235. C2×#41C2×D37semidirect
236. C22×Q8C2×D37semidirect
255. #124C2×D37semidirect
257. C22⋊Q8C2×D37semidirect
262. #121C2×D37semidirect
265. C4×Q8C2×D37semidirect
359. Q8D4×D37semidirect
416. C22#11924561semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#119245612
C22×#105969412
D4×D1482
#145579232
#145577812
#145577672
#145577452
#145577332
#145576832
#145576782
#145576722
#145576662
#145576312
#145576052
C37×#5482
#326537